Calhoun: The NPS Institutional Archive

Reports and Technical Reports All Technical Reports Collection

2003

Policies for biodefense revisited: the

prioritized vaccination process for Smallpox

Kress, Moshe

Monterey, California. Naval Postgraduate School

http://hdl.handle.net/10945/737

brought to you by COREView metadata, citation and similar papers at core.ac.uk

provided by Calhoun, Institutional Archive of the Naval Postgraduate School

NPS-OR-03-008

NAVAL POSTGRADUATE SCHOOL Monterey, California

Policies for Biodefense Revisited: The Prioritized Vaccination Process for

Smallpox

by

Moshe Kress

September 2003

Approved for public release; distribution is unlimited. Prepared for: Naval Postgraduate School

Monterey, California 93943-5000

2

NAVAL POSTGRADUATE SCHOOL MONTEREY, CA 93943-5000

RADM David R. Ellison Richard Elster Superintendent Provost This report was prepared for and funded by the Naval Postgraduate School, Monterey, CA 93943-5000. Reproduction of all or part of this report is authorized. This report was prepared by: MOSHE KRESS Professor of Operations Research Reviewed by: R. KEVIN WOOD Associate Chairman for Research Department of Operations Research Released by: JAMES N. EAGLE LEONARD A. FERRARI, Ph.D. Chairman Associate Provost and Dean of Research Department of Operations Research

3

REPORT DOCUMENTATION PAGE Form Approved OMB No. 0704-0188 Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instruction, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188) Washington, DC 20503. 1. AGENCY USE ONLY (Leave blank)

2. REPORT DATE September 2003

3. REPORT TYPE AND DATES COVERED Technical Report

4. TITLE AND SUBTITLE: Policies for Biodefense Revisited: The Prioritized Vaccination Process for Smallpox 6. AUTHOR(S) Moshe Kress

5. FUNDING NUMBERS

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Naval Postgraduate School Monterey, CA 93943-5000

8. PERFORMING ORGANIZATION REPORT NUMBER NPS-OR-03-008

9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) N/A

10. SPONSORING / MONITORING AGENCY REPORT NUMBER

N/A

11. SUPPLEMENTARY NOTES The views expressed in this thesis are those of the author and do not reflect the official policy or position of the Department of Defense or the U.S. Government. 12a. DISTRIBUTION / AVAILABILITY STATEMENT Approved for public release; distribution is unlimited.

12b. DISTRIBUTION CODE

13. ABSTRACT (maximum 200 words) Handling bioterror events that involve contagious agents is a major concern in the war against terror, and is a cause for

debate among policymakers about the best response policy. At the core of this debate stands the question which of the two post-event policies to adopt: mass vaccination—where maximum vaccination capacity is utilized to uniformly inoculate the entire population, or trace (also called ring or targeted) vaccination—where mass vaccination capabilities are traded off with tracing capabilities to selectively inoculate only contacts (or suspected contacts) of infective individuals. We present a dynamic epidemic-intervention model that expands previous models by capturing some additional key features of the situation and by generalizing some assumptions regarding the probability distributions of inter-temporal parameters. The model comprises a set of difference equations. The model is implemented to analyze alternative response policies. It is shown that a mixture of mass and trace vaccination policies—the prioritized vaccination policy—is more effective than either of the two aforementioned policies.

15. NUMBER OF PAGES

30

14. SUBJECT TERMS Mass Vaccination, Trace Vaccination, Prioritized Vaccination, Potentially Traceable

16. PRICE CODE

17. SECURITY CLASSIFICATION OF REPORT

Unclassified

18. SECURITY CLASSIFICATION OF THIS PAGE

Unclassified

19. SECURITY CLASSIFICATION OF ABSTRACT

Unclassified

20. LIMITATION OF ABSTRACT

UL

ABSTRACT

Handling bioterror events that involve contagious agents is a major concern in the war against terror, and is a cause for debate among policymakers about the best response policy. At the core of this debate stands the question which of the two post-event policies to adopt: mass vaccination—where maximum vaccination capacity is utilized to uniformly inoculate the entire population, or trace (also called ring or targeted) vaccination—where mass vaccination capabilities are traded off with tracing capabilities to selectively inoculate only contacts (or suspected contacts) of infective individuals. We present a dynamic epidemic-intervention model that expands previous models by capturing some additional key features of the situation and by generalizing some assumptions regarding the probability distributions of inter-temporal parameters. The model comprises a set of difference equations. The model is implemented to analyze alternative response policies. It is shown that a mixture of mass and trace vaccination policies—the prioritized vaccination policy—is more effective than either of the two aforementioned policies.

1

1. Introduction

Responding to a bioterror event that involves contagious agents has become a major

issue in the war against terror. There are many operational and logistic decisions that

must be made carefully in order to effectively cope with such a threat. These decisions

are roughly divided into two levels: Structural (strategic) decisions that need to be made

in advance, and operational (real-time) decisions that must be made during the event.

Some of the structural (strategic) problems are:

• How many vaccines to produce and stock?

• Which supply management policies to apply for allocating, deploying, and

controlling inventories of vaccines and other related supplies?

• What infrastructure (vaccination stations, quarantining facilities, etc.) is required?

• What vaccination procedure (e.g., inoculation only, pre-vaccination screening for

contra indication) to adopt?

• How to determine the manpower requirements and personnel assignment?

The operational (real-time) decisions include:

• Identifying the type of the bioterror event.

• Managing the contact tracing process (if applied).

• Prioritizing efforts with respect to monitoring, isolating, quarantining, tracing, and

vaccinating.

• Coordinating the supply chain of vaccines and other supplies.

• Identifying bottlenecks and potential congestion.

• Determining capacities and setting service rates.

One of the most critical decisions—a decision that has both structural and operational

implications—is which vaccination policy to adopt. This question has generated debate

among policymakers [1] and has also drawn much attention by the general public [2], [3].

The vaccination policy decision has two levels. At the first level, policymakers must

choose between essentially two options: a preemptive approach in which the entire

population is pre-vaccinated, and a “wait and see” approach where post-attack emergency

2

response (vaccination, quarantine, isolation) commences following an outbreak of the

disease. Mixtures of these two options are possible too, i.e., pre-vaccination of first

responders (e.g., health-care and law-enforcement personnel) only. Sociological and

psychological considerations (is there a real threat or just a perceived one?) coupled with

medical considerations (fear of side effects) have hindered policy makers from taking any

significant preemptive action so far.

If no significant preemptive measures are taken, the question at the second level is

which post-event vaccination policy to adopt. The two policies that have been examined

so far are mass vaccination and trace vaccination. In mass vaccination, maximum

vaccination capacity is utilized to uniformly inoculate the entire population. In trace (also

called ring or targeted) vaccination, only limited vaccination capacity is utilized to

selectively inoculate contacts (or suspected contacts) of infective individuals.

Several researchers have attempted to address the issue of the post-event vaccination

process in the case of smallpox, and in particular to compare mass vaccination to trace

vaccination. Kaplan, Craft, and Wein [4] propose a continuous-time deterministic model

that comprises 17 ordinary differential equations. Their model, details of which are

reported in [5], captures many important aspects of the situation, including the “race to

trace.” The race to trace reflects the time constraints on the effectiveness of the

vaccination process due to the limited time period in which an infective is

vaccine-sensitive or “immunable.” They assume exponential distributions with regard to

all of the time parameters (e.g., incubation time, infectious time) and therefore the

transitions in their model are not dependent on the “age” of an individual in a certain

stage of the epidemic. They also ignore the effect of the epidemic initial conditions. Some

epidemic and operational parameters may have different values at the early stages of the

epidemic than later on. For example, the vaccination process may need some setup time

during which only a portion of the potential vaccination capacity can be utilized. Also,

during the first generation of the disease (prior to detection) the infection rate may be

higher and the isolation rate may be lower because of lack of situational awareness.

Kaplan et al. [4] conclude that under reasonable conditions regarding the initial attack

3

size and the epidemic’s spread parameters, mass vaccination is generally more effective

than trace vaccination.

Contrary to the analytic macroscopic approach in [4], [5], Halloran, Longini, Azhar,

and Yang [6] use a detailed simulation to model at the micro level a smallpox

transmission process in a (socially) structured community of 2,000 people. The disease

transmission process in their model takes into account the social structure of the

community, attempting to better represent the way the epidemic spreads in the

population. They assume that the random variables that are associated with the duration

of the various stages of the epidemic are uniformly distributed—an assumption that can

hardly be justified. They assert that according to their simulation, trace vaccination would

prevent more smallpox cases per dose of vaccine than would mass vaccination. Due to

the imbedded nonlinearities in the epidemic process, it is not clear if their conclusions

derived for a population of 2,000 may also apply to a population of say 10 million. Their

model also lacks the operational and logistical aspects that are accounted to in [4].

However, the Halloran et al. and the Kaplan et al. models are not inconsistent. The model

in [4] gives similar results as the model in [6] when supplied the inputs used in [6]

(see [7]).

A Markov chain model of the epidemic progression is utilized by Meltzer, Damon,

LeDuc, and Millar [8] to analyze various response options. Their conclusion is that only a

combination of vaccination with an effective quarantine may eradicate the epidemic. The

paper contains some epidemic progression data—some of which is used in our paper.

Koopman [9] reviews the studies the models in [4] and [6] and suggests a possible third

modeling approach based on a network model that describes the links among individuals.

Such models are reported in [10] and [11]. Other researchers ([12], [13]) use

distance-based models to analyze ring vaccination, which is a geographically oriented

version of trace vaccination. Spatial effects are also examined in [14], where a high-

resolution computational model is developed. In a more recent publication [15], the

authors develop a stochastic model of outcomes under various control policies. Their

4

model is limited in the sense that they assume, rather than derive, the post intervention

basic reproductive rate.

There are two main objectives in the current research. The first objective is to develop

a flexible, large-scale analytic model that expands and generalizes previous models. Our

model is conceptually similar to the model in [4], but it differs in the use of discrete

rather than continuous time. Thus, it is in the form of a set of difference equations rather

than differential equations. As it will be shown later on, a discrete model can more easily

capture certain key operational, logistical, and epidemiological aspects of the situation.

Also, in contrast to the constant hazard functions (exponential distributions) of the time

parameters in [4] and the uniform distributions of these parameters in [6], our model

makes no assumptions with regard to these distributions. It can take any finite-support

distributions—including empirical. The model also explicitly represents the

age-dependent transitions among the various stages of the epidemic. We also distinguish

between the initial conditions of the epidemic and its operational and logistical

steady-state parameters.

The second objective is to propose an alternative vaccination policy, the prioritized

vaccination policy (PVP), which may be viewed as a mixture of the mass vaccination

policy (MVP) and the trace vaccination policy (TVP). We demonstrate that under a set of

realistic assumptions regarding the epidemic parameters and the operational and logistical

capabilities to handle it, the PVP is significantly more effective than either the MVP or

the TVP individually.

The paper is organized as follows: in Section 2 we discuss the possible response

options for a bioattack, and in Section 3 we present the model. The three vaccination

policies—PVP, MVP, and TVP—are analyzed in Section 4. First, we examine the base

case, which is similar to the scenario described in [4], and then we perform sensitivity

analysis. Section 5 contains the summary and conclusions.

5

2. The Epidemic and the Possible Interventions

We consider a situation where a malevolent agent engages a population with an act of

terror by releasing the smallpox virus in a public area. This act of terror is clandestine,

and the authorities are not aware of the event until a certain number of symptomatic

patients are reported and diagnosed as carrying the disease. Once the epidemic is detected

and identified, a response is initiated, which involves isolation, quarantine, tracing

contacts and vaccinating some or all of the population. The disease has an incubation

period before an infected individual becomes symptomatic. During the incubation period

the infected individuals are not infectious, and therefore the disease is not transmitted to

others. The incubation period is divided into two periods of time: the immunable (also

called vaccine sensitive) period and the non-immunable period. During the immunable

period, vaccination is effective. It will eradicate the disease from an infective at that stage

with high probability. During the non-immunable period, vaccination is not effective, and

therefore the infective will eventually become ill. Once the incubation period is over, the

infected individual becomes infectious. The infectious period lasts as long as the

symptoms still persist. As in [4], the transmission of the disease is in the form of

homogeneous mixing.

At any given time t, the population of non-vaccinated individuals is divided into the

following six possible stages:

S: Susceptible to the disease;

A: Infected, not yet infectious (incubating), and immunable (vaccine sensitive);

B: Infected, not yet infectious (incubating), and not immunable;

I: Infected, infectious, and not yet isolated;

Q: Infected, infectious, and isolated,

R: Removed, recovered, and immune or dead.

The durations of the stages A, B, I, and Q are random variables with probability mass

functions ( ), ( ), ( ), and ( ),A B I QP j P j P j P j respectively. The parameter j indicates the

6

number of days—the time resolution of the model—an individual stays at a certain stage.

Note that while ( ), ( ), and ( )A B QP j P j P j are determined purely by epidemic

characteristics, PI(j) is affected also by the response process and in particular, the

effectiveness of the detection and quarantine efforts. Numerical values of the various

probability mass functions that are selected for the base case in the analysis are taken

from [8].

Once the epidemic is identified, detected infectious individuals (in stage I) are

isolated (moved to stage Q) and the vaccination process commences. Two potential

vaccination queues may be formed: the general queue and the tracing queue. The general

queue comprises non-vaccinated individuals (in stages S, A, and B) who are not in the

tracing queue. The tracing queue comprises all non-vaccinated contacts named by an

index case. An index case is a newly detected infectious individual. Because of the

additional effort that is required to trace contacts, the service rate at the tracing queue is

lower than that at the general queue. The tracing service reduction factor is the ratio

between the service rates at the general queue and tracing queue. We assume that a

certain portion of the vaccination capacity is allocated to the tracing queue and the rest is

applied to the general queue.

The set of contacts that are named by a certain index case is called the index set. Let

E denote the set of index cases and let Ji denote the index set of i E∈ . The index set Ji

may comprise three possible disjoint subsets:

Ji(1) – Infected individuals not yet vaccinated;

Ji(2) – Non-infected (susceptible) individuals not yet named or vaccinated;

Ji(3) – Individuals named and vaccinated earlier by another index or already vaccinated

in the general queue.

For a given index i the target population of the tracing process is Ji(1). Tracing

individuals in Ji(2) is somewhat wasteful since there is no race to trace. The susceptible

7

individuals in Ji(2) can be vaccinated in the more efficient general queue. Tracing

individuals in Ji(3) is clearly a waste of tracing capacity. Thus,

(1) (2) (3)i i i iJ J J J= U U (1)

and the tracing queue is generated by ii E

J∈U .

To properly represent the effective vaccination rate at the tracing queue we need to

introduce three terms: (index case) i-infective, newly named i-infective, and potentially

traceable. An i-infective is an individual that has been infected by index case i. For each

infective the transmitter is uniquely defined. That is, an infective cannot be infected

twice. Notice that an i-infective may be named by another non-transmitter index case j

before being named by its transmitter i. An i-infective, not previously named by another

index case, nor vaccinated, which is named by index case i is called newly-named

i-infective. An i-infective, not yet traced or vaccinated, is said to be potentially traceable

(PT) if its corresponding transmitter i has been detected and interviewed. Let Pi denote

the set of PT i-infectives. Clearly, the sets Pi, i E∈ are disjoint. A newly named

i-infective is PT, but a PT i-infective may not be newly named if the corresponding index

case failed to name this contact. The state PT is transient, that is, an i-infective is PT only

during the time period when i is interviewed. An i-infective that has not been named by

its corresponding index i becomes non-PT once again in the next period. Thus, an

i-infective may pass through three possible states: pre-PT, PT, and post-PT. By

definition, an individual in the post-PT state will never become PT again. He may be

traced however by a non-transmitter index case. Also, an i-infective in the pre-PT state

may never become PT if named by a non-transmitter. As in [4], we assume that

iJ M= for all index case i. Also, since we assume homogeneous mixing, that is, all

index cases are identical with respect to the transmission process, iP N= for all index

cases i E∈ . However, unlike M that stays constant throughout the epidemic

(e.g., M = 50 in [4]), N = N(t) changes over time as the epidemic progresses. As in [4], we

assume that M ≥ N(t) for all t, that is, the size of the index set always exceeds the number

8

of PT infectives. Finally, let ω0 denote the probability that a PT infective is (newly)

named by the corresponding index.

Note that the total number of PT infectives at time t is Q1(t)N(t), where Q1(t) is the

number of index cases that are detected and interviewed at time t. Similarly, Q1(t)M is the

total number of names generated at time t by the index cases. We assume that M

represents the net number of newly named contacts. Therefore, the index sets are disjoint.

Let ( )tω denote the fraction of the index set that contains newly named infectives.

0( )( ) .N tt

Mω ω= (2)

Since there are no reliable estimates for 0ω , we will first assign, in the base case, a

reasonable value for this parameter, and then we will perform a full-scale sensitivity

analysis in Section 4.

The definitions in (2) are used as a backdrop for the more generalized definitions in

Section 3.

Note that MVP and TVP generate only one queue each—general queue in MVP and

tracing queue in TVP. The vaccination process that is proposed in this paper—PVP,

which is described in the next section—generates both queues.

Figure 2.1 depicts the epidemic stages and the transitions among them. The subscripts

indicate the age of a certain stage. The terms 0, ,j j jA A A− + denote cohorts at epidemic age j

in the pre-PT, PT, and post-PT stages, respectively. Notice that in the absence of a

corresponding index case, an individual may move from stage A− to stage B without

being potentially traceable at all (see e.g., edge 1 2A B− → ). One of the main objectives is

to trace, as fast as possible and as many as possible, individuals in stage 0A (see the

shaded oval shape in Figure 2.1).

9

Figure 2.1: The Epidemic Stages

3. The Model

3.1 Notation

The various cohorts in the epidemic are:

0 0

( ) Number of susceptibles at time ;( ) Number of infectives in the -th day of the incubation period,

( ) ( ) ( ) ( ), where ( ) are pre- , ( ) are and (j

j j j j j j j

S t tA t immunable j

A t A t A t A t A t PT A t PT A t− + − +

−−

= + +

0 0

) are post- ;

( ) Number of - infectives in the -th day of the incubation period,

( ) ( ) ( ) ( ), where ( ) are pre- , ( ) are and ( ) are post- ;

( ) Number of in

j

j j j j j j j

j

PT

B t non immunable j

B t B t B t B t B t PT B t PT B t PT

I t

− + − +

−

= + +

− fectious individuals in the -th day of the infectious period;

( ) Number of isolated individuals in the -th day of isolation.j

j

Q t j−

.

.

Susceptibles Incubating Incubating Infectious Isolated RemovedImmunable Non-immunable

Pre-PT PT Post-PT

A3-

A2-

A1-

B3

B2

B6

B5

B4

I1

I4

I3

I2 Q2

Q1

Q5

Q4

Q3

S

A30

A20

R

A10

.

.

.

.

A4+

A3+

A2+

.

.

.

...

.

.

Susceptibles Incubating Incubating Infectious Isolated RemovedImmunable Non-immunable

Pre-PT PT Post-PT

A3-

A2-

A1-

B3

B2

B6

B5

B4

I1

I4

I3

I2 Q2

Q1

Q5

Q4

Q3

S

A30

A20

R

A10

.

.

.

.

A4+

A3+

A2+

.

.

.

...

10

Let,

0 0

1 1 1

0 0

1 1 1

1

( ) ( ), ( ) ( ), ( ) ( ),

( ) ( ), ( ) ( ), ( ) ( ),

( ) ( ).

j j jj j j

j j jj j j

jj

A t A t A t A t A t A t

B t B t B t B t B t B t

I t I t

∞ ∞ ∞− − + +

= = =

∞ ∞ ∞− − + +

= = =

∞

=

= = =

= = =

=

∑ ∑ ∑

∑ ∑ ∑

∑

(3)

Other parameters are:

( )tα – Infection rate;

T(t) – The length of the tracing queue;

L(t) – Total number of newly named infectives in the tracing queue;

( )Tr t – Tracing rate;

V(t) – Nominal vaccination/tracing capacity;

q – Proportion of the vaccination capacity allocated to the tracing queue;

( )Gr t – Vaccination rate in the general queue;

r+(t) – The residual trace vaccination rate that is applied to individuals who are not

newly named PT infectives (Recall that a newly-named PT infective is an

infected individual that is named, for the first time, by the corresponding

transmitter);

( )tω – Estimated fraction of the tracing queue that contains newly named infectives;

0ω – The probability of naming a PT infective;

w(t) – The rate at which non-PT infectives become PT;

M – Size (cardinality) of the index set (average number of traced individuals per

index case);

c – Tracing service reduction factor;

e – Vaccination efficacy (% of vaccinations that result in a “take”). To simplify the

exposition we assume initially that e = 1. This assumption is relaxed in the

analysis; and

D – Epidemic detection threshold.

11

3.2 Definitions and Derivations of Transitions

The infection rate is given by

0 ( )( ) ,

(0)R E ItS

α = (4)

where R0 is the basic reproductive ratio, E(I) is the mean duration of the infectious stage,

and S(0) is the size of the population at the beginning of the epidemic.

The tracing rate is

{ ( ), ( )} if ( ) 0( )( ) .

0 OtherwiseT

Min qV t cT t T tcT tr t

>=

(5)

The tracing queue is given by the following recursive equation:

1( ) ( 1)(1 ( 1)) ( ),TT t T t r t MQ t= − − − + (6)

where Q1(t) is the number of new index cases at time t.

The number of newly named infectives in the tracing queue is given recursively by

0 0

0( ) ( 1)(1 ( 1)) ( ( ) ( )),TL t L t r t A t B tω= − − − + + (7)

The sum 0 0( ) ( )A t B t+ is the total number of PT infectives at time t.

The vaccination rate in the general queue is,

0 00

( ) ( ) ( )( ) 1, .

( ) ( ) ( ) ( ) ( ) (1 )( ( ) ( ))T

GV t cT t r t

r t MinS t A t A t B t B t A t B tω− + − +

−=

+ + + + + − + (8)

12

Note that the general queue may include individuals that are susceptible, non-PT

(both immunable and non-immunable), and PT not newly named.

Let,

( )( ) .( )

L ttT t

ω = (9)

( )tω is the estimated fraction of the tracing capacity that is applied to newly named

PT infectives. It can be seen that (9) is a natural generalization of (2) for the tracing

queue. The remaining portion 1- ( )tω of the tracing capacity generates the residual trace

vaccination rate, which is applied proportionally to individuals that are not newly named

infectives. Thus,

0 00

(1 ( )) ( ) ( )( ) .

( ) ( ) ( ) ( ) ( ) (1 )( ( ) ( ))Tt T t r t

r tS t A t A t B t B t A t B t

ωω

+− + − +

−=

+ + + + + − + (10)

The vaccination rates are non-zero only after the epidemic is detected. Detection

occurs only after D symptomatic individuals report to emergency rooms. D is the

epidemic detection threshold, which indicates the alertness and responsiveness of the

medical and public health system.

The hazard functions of the various stages are:

1

0

( )( )1 ( )

XX j

Xk

P jjP k

ϕ −

=

=−∑

, X=A, B, I, Q. (11)

The rate at which pre-PT infectives become PT depends on the rate at which

infectious individual become index cases.

13

Let

0

( ) , 1, 2,....1 ( )

l Ij j

Ik

P j l lP k

β

=

+= =

−∑ (12)

ljβ is the probability that an infectious individual, who is currently at the j-th day of

his/her infectious period (I), will be detected l days from now. Notice that if l = 1, then 1 ( )j B jβ ϕ= .

Based on our assumption of homogeneous free mixing, the probability ul(t) that an

individual who got infected on day t will become PT on day t+l is,

1

1

( )( ) .

( )

lk k

kl

kk

I tu t

I t

β∞

=∞

=

=∑

∑ (13)

Thus, the probability wj(t) that a pre-PT infective who is at the j-th day of the

incubation period at day t, will become PT at t+1 is,

1

1

( )( ) .

1 ( )

jj j

ll

u t jw t

u t j

+

=

−=

− −∑ (14)

3.3 The Difference Equations

The following set of difference equations describes the epidemic progression in case

of intervention:

( 1) ( )[1 ( ) ( )][1 ( ) ( )]GS t S t t I t r t r tα ++ = − − − (15)

1 1( 1) [ ( ) ( ) ( )][1 ( ) ( )](1 (t))GA t t S t I t r t r t uα− ++ = − − − (16)

14

01 1( 1) [ ( ) ( ) ( )][1 ( ) ( )] (t)GA t t S t I t r t r t uα ++ = − − (17)

1( 1) ( )[1 ( ) ( )](1 ( ))(1 (t))j j G A jA t A t r t r t j wϕ− − ++ + = − − − − (18)

01( 1) ( )[1 ( ) ( )](1 ( )) (t)j j G A jA t A t r t r t j wϕ− ++ + = − − − (19)

1

00 0

( 1) ( ( )(1 ( ) ( ))

( )(1 ( ) (1 )( ( ) ( ))))(1 ( ))j j G

j T G A

A t A t r t r t

A t r t r t r t jω ω ϕ

+ + ++

+

+ = − +

+ − − − + − (20)

1( 1) ( ( )(1 ( )) ( ) ( ))(1 ( ) ( ))(1 (t))j j B j A G jB t B t j A t j r t r t wϕ ϕ− − − ++ + = − + − − − (21)

01( 1) ( ( )(1 ( )) ( ) ( ))(1 ( ) ( )) (t)j j B j A G jB t B t j A t j r t r t wϕ ϕ− − ++ + = − + − − (22)

1

0 00 0

( 1) ( ( )(1 ( )) ( ) ( ))(1 ( ) ( ))

( ( )(1 ( )) ( ) ( ))(1 ( ) (1 )( ( ) ( )))j j B j A G

j B j A T G

B t B t j A t j r t r t

B t j A t j r t r t r t

ϕ ϕ

ϕ ϕ ω ω

+ + + ++

+

+ = − + − − +

+ − + − − − + (23)

*1

0 *0 0

( 1) (( ( ) ( ))(1 ( ) ( ))

( )(1 ( ) (1 )( ( ) ( )))) ( ) ( )(1 ( ))j j j G

j T G A j B

B t A t A t r t r t

A t r t r t r t j B t jω ω ϕ ϕ

− + ++

+

+ = + − − +

+ − − − + + − (24)

*1

1

( 1) ( ) ( )j Bj

I t B t jϕ∞

=

+ =∑ (25)

1( 1) ( )(1 ( ))j j II t I t jϕ+ + = − (26)

11

( 1) ( ) ( )j Ij

Q t I t jϕ∞

=

+ =∑ (27)

1( 1) ( )(1 ( ))j j QQ t Q t jϕ+ + = − (28)

Explanation of the Equations

First we observe that the vaccination rates are as follows: a fraction 0ω of the PT

infectives—the newly named infectives—are vaccinated at a rate rt(t), while the rest of

the population is vaccinated at a rate rG(t) + r+(t). Recall that the parameter ( )r t+ is the

residual portion 1- ( )tω of the trace vaccination capacity that is applied to (“wasted” on)

individuals that are not newly named PT infectives.

Equation (15): The remaining susceptibles are those who have not been vaccinated nor

infected.

15

Equations (16), (17): The newly infected are among those who have not been vaccinated

neither in the general queue nor by the residual tracing capacity. The parameter u1(t) is

the probability that a newly infected becomes immediately PT.

Equation (18): The immunable pre-PT infectives are those who have not been

vaccinated (neither in the general queue nor by the residual tracing capacity), are still

immunable (1 ( )A jϕ− ) and have not become PT (1-wj(t)).

Equation (19): The immunable PT infectives at time t+1 comprise immunable pre-PT

infectives ( ( )jA t− ) who have not been vaccinated, are still immunable (1 ( )A jϕ− ), and

have become PT (wj(t)).

Equation (20): The immunable post-PT infectives at time t+1 comprise previously

immunable PT ( 0 ( )jA t ) and post-PT ( ( )jA t+ ) infectives that have not been vaccinated and

remain immunable (1 ( )A jϕ− ). The vaccination rate of the PT individuals is a convex

combination of the (net) tracing rate and the combined vaccination rate of the general

queue and the residual tracing capacity.

Equations (21)-(23): These equations are similar to (18)-(20). They represent the

transition from an immunable stage to a non-immunable stage.

Equations (24): This equation records the total number of the non-immunable infectives.

We need the two representations of stage B cohort ((21)-(23) and (24)) because of the

fact that vaccinating individuals at that stage is ineffective; they will eventually become

sick. Thus, (21)-(23) are needed for determining the vaccination queue sizes for pre-PT,

PT, and post-PT infectives, while (24) counts the individuals who will eventually become

infectious (see Equation (25)).

Equation (25): The newly infectious individuals comprise infectives whose incubation

period has ended.

16

Equation (26): The remaining infectious individuals at stage I are those who have not

been detected yet.

Equation (27): The newly isolated infectious individuals (new index cases) are those

who have been detected.

Equation (28): The remaining individuals in isolation are those who have not been

removed yet (recovery or death).

The model that has been described above is general and can represent several

vaccination policies. Specifically, once the vaccination process has been initiated, the

MVP implies that 0( ) ( ) 0 for all , and ( ) 0T Gr t r t t r tω+= = = > for all t, such that S(t) > 0.

In TVP, ( ) 0 for all , and ( ), ( ) 0G Tr t t r t r t+= > for all t, such that T(t) > 0.

The PVP policy is a combination of mass vaccination and trace vaccination where at

all times treating the tracing queue preempts the general queue. Whenever there are new

index cases, an appropriate vaccination capacity is allocated for tracing and vaccinating

the generated named contacts (Index Set). The tracing/vaccination capacity allocated to

the tracing queue is limited only by the total existing vaccination capacity, that is, q = 1.

Mass vaccination is carried on with the remaining vaccination capacity. The parameter c,

the tracing service reduction factor, quantifies the inefficiencies that result from the

tracing process.

4. Analysis

The model developed in Section 3 is applied now to evaluate the effectiveness of the

prioritized vaccination process (PVP) in comparison with the mass vaccination process

(MVP) and the trace vaccination process (TVP). Recall that in PVP the first priority is for

the tracing queue, and the remaining vaccination capacity is applied to the general queue.

An individual is treated in the general queue only if he is not claimed by the tracing

queue.

17

4.1 Base Case Whenever it is relevant, the parameters chosen for the base case are similar to those in

[4]. We also assume for this case that the values of the parameters remain constant

throughout the duration of the epidemic. In particular, there are no special initial

conditions. The values of the various epidemic, population, and operational parameters

are shown in Table 4.1. Table 4.2 presents the characteristics of the probability mass

functions of the time parameters. These probability distributions are consistent with the

assumptions in [4] and the data in [8].

Value Definition Symbol 500,000* Daily vaccination capacity V(t)

10-7 Infection rate α(t) 1,000 Number of initially infected

1 (0)A− 107 Size of population S(0) 4 Tracing service reduction factor c 50 Size of the index set M 0.7 Probability of naming a PT infective ω0 20 Epidemic detection threshold D

0.975 Vaccination efficacy E * In [4] the vaccination capacity is assumed to be 106. We believe that 500,000 is a more realistic estimate, at least for Israel.

Table 4.1: The Base Case Parameters

Time Period P0.05 P0.95 Mean MedianIncubation (A) 1 5 3 3 Incubation (B)* 8 14 11.5 11.5

Infectious (I) 1 5 3 3 Isolation (Q) 9 15 12 12

* Time is measured from the day of infection.

Table 4.2: The Base Case Probability Distributions.

The probability distributions in Table 4.2 are also consistent with the limited data

regarding the probable durations of the various stages that are reported mainly in [16] and

[17].

18

The results corresponding to the base case are summarized in Table 4.3:

Vaccination

Policy Number Infected*

Duration of Epidemic

Maximum Daily Isolation Capacity

Needed PVP 1,015 57 days 1,002 beds MVP 2,232 58 days 1,765 beds TVP > 108,000 > 1 year 9,380 beds

* Excluding the initially infected (1,000).

Table 4.3: Results for the Base Case.

If we take the exact same parameters as in [4], that is, V(t) = 106 and ω0 = 0.5, then

the numbers of infected are 560 and 818 for PVP and MVP, respectively.

Figure 4.1 depicts the progression of the epidemic under each one of the three

vaccination policies. The graphs indicate the number of newly infected.

0

50

100

150

200

250

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43

days

# in

fect

ed PVP

MVP

TVP

Note: Including the initially infected that are depicted in the graph of days 6-16.

Figure 4.1: The Epidemic Progression.

Figure 4.1 demonstrates the significant differences among the three vaccination

policies: MVP results in a relatively large number of second-wave infected individuals,

but the epidemic is eradicated much faster than when TVP is executed. In TVP, the

epidemic has initially smaller peaks than in MVP, but it is expanding gradually over a

longer period of time (in the base case, the epidemic reaches its peak on day 159). In

19

PVP, the epidemic has smaller peaks than in TVP and its duration is about the same as in

MVP.

Discussion

The PVP is clearly more effective than the two other vaccination policies in the base

case. It results in over 50% less derived infections (second wave and later infections that

result from the initial attack of 1,000 infected) compared to MVP, and over 99% less

derived infections compared to TVP. In both PVP and MVP, the epidemic is eradicated

(and the entire population is inoculated) after almost two months. In the TVP case, the

epidemic lasts a little over one year. The peak isolation/quarantine capacity needed for

the PVP is less than 60% the peak capacity needed for MVP. The extra capacity needed

in the MVP case is required relatively early in the epidemic since the first waves of

infection may not be captured by the untargeted homogeneous vaccination process. In

TVP and PVP, the vaccination is targeted at high-risk susceptibles, and therefore more

stage-A individuals may be located and vaccinated at the early stages of the epidemic.

However, the lack of massive vaccination in TVP results in prolonging and expanding the

epidemic and thus increasing the total number infected. The peak isolation capacity in the

TVP case (9,380) is needed on days 165 and 166 of the epidemic. These results are

consistent with the conclusions in Kaplan et al. that MVP is significantly more effective

than TVP.

Note that the relative high efficiency of PVP is due to the synergy that is created by

combining mass and trace vaccination. Mass vaccination builds up herd immunity that

effectively reduces the value of R0, thus amplifying the effect of the tracing part of the

process (See Figure 3 in [4]).

In the absence of effective disease detection systems that can indicate the occurrence

of a bio-attack event, the infected individuals in the first wave of infection—those who

were infected by the initial bio-attack—cannot be helped. That is why the three graphs in

Figure 4.1 coincide during days 1-16.

20

4.2 Sensitivity Analyses Is the evident dominance of PVP over MVP and TVP robust? Does it dominate for

other sets of parameters? To address these questions we perform sensitivity analysis with respect to some key

parameters. Since in most cases TVP turns out to be inferior to PVP and MVP by almost

two orders of magnitude, the comparisons in the following will focus at MVP and PVP

only. The number of infectives shown in the analysis excludes those who were initially

infected by the bioattack ( 1 (0)A− ).

Tracing Service Reduction Factor – c

In the base case, we assumed that the tracing process consumes four times the

vaccination resources needed for mass vaccination, that is, c = 4. This value is adopted

from [4]. Arguably, the larger the value of c, the lower the relative efficiency of the

tracing process, and thus the less likely it is that PVP will outperform MVP. Figure 4.2

presents the effect of increasing c in increments of 10 on the effectiveness of PVP.

Obviously, the varying of c does not affect the MVP, since there is no tracing activity.

For PVP, the number of infected increases from 990 (c = 1). The break-even value of the

tracing service Reduction Factor for PVP compared to MVP is c = 65. In other words, the

tracing service rate must be more than 65 times slower than the general vaccination rate

to render MVP more effective.

0

500

1000

1500

2000

2500

3000

1 10 20 30 40 50 60 70

c

# in

fect

ed

PVP

MVP

Figure 4.2: Number of Infected as a Function of the Tracing Service Reduction Factor.

21

Size of the Index Set – M

Similarly to the tracing service reduction factor c, the average number of cases that

are traced per index case M does not affect MVP. However, increasing M, without

increasing the value of ω0 at the same time, will clearly have a negative effect on PVP

(and TVP) because of the inefficiencies that result from the tracing service reduction

factor. Figure 4.3 presents the effect of varying M between 50 (the base case) and 500 on

the performance of PVP. Notice that the effect is relatively small in this range. Increasing

M by an order of magnitude results in less than 35% more infectives.

0

500

1000

1500

2000

2500

50 100 150 200 250 300 350 400 450 500

M

# in

fect

ed

PVPMVP

Figure 4.3: Number of Infected in PVP as a Function of the Size of the Index Set.

Epidemic Detection Threshold – D

One would expect that the effectiveness of the response process will depend on the

situational awareness of the healthcare system. The faster the outbreak of the epidemic is

detected, the earlier response actions can be taken, and therefore fewer infected cases

would be expected. Figure 4.4 shows the effect of the epidemic detection threshold on

PVP and MVP. The effect of varying the detection threshold in the PVP case is similar to

the effect in the MVP case: Moving from D = 100 to a fully alert system (D = 1) results in

33% less infectives in both cases.

22

0

500

1000

1500

2000

2500

3000

3500

1 20 50 100

D

# in

fect

ed PVPMVP

Figure 4.4: Number of Infected as a Function of the Detection Threshold.

Daily Nominal Vaccination Capacity – V

The results are sensitive to the assumption regarding the effective daily vaccination

capacity, which is assumed here to be fixed over time. Figure 4.5 presents the effect of

varying the values of V.

0

2000

4000

6000

8000

10000

12000

14000

200K 300K 400K 500K 600K 700K 800K 900K 1M

Vaccination Capacity V

# In

fect

ed

PVPMVP

Figure 4.5: Number of Infected as a Function of the Vaccination Capacity.

Clearly, the effectiveness of MVP is more sensitive to the value of V than PVP. The

advantage of PVP over MVP is most notable for small values of V. When the vaccination

capacity is only 100,000 a day (not shown in the graph), the number of infected in the

MVP case is more than 14 times higher than that number in the PVP case. This advantage

23

declines as V increases since the relative impact of the tracing process gets smaller. If

V = 1M, then PVP results in 45% fewer casualties compared to MVP.

Tracing Effectiveness – ω0

This parameter reflects the efficiency of the tracing process. It is the maximum

possible probability to trace an infective. In the base case, we assumed that this efficiency

cannot exceed 70%. Clearly, the lower the upper bound, the less advantageous are PVP

and TVP compared to MVP. Figure 4.6 shows the effect of ω0 on the number of

infectives. TVP is more effective than MVP if ω0 ≥ 0.99, that is, only in the case of

extremely high tracing effectiveness. For ω0 ≥ 0.1, PVP is the most effective policy. Its

advantage over MVP increases, as ω0 gets larger. If ω0 = 0.5, then PVP results in 39%

fewer infectives compared to MVP. If ω0 = 0.9, then the number of infectives is 70%

lower.

0

500

1000

1500

2000

2500

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Tracing Effectiveness

# In

fect

ed

PVP

MVP

Figure 4.6: Number of Infected as a Function of ω0. Effect of Initial Conditions

During the initial stages of the epidemic some of the parameters may have different

values than later on in the epidemic. We consider here three such parameters: the daily

vaccination capacity V, the infection rate α, and the distribution function of the duration

of the infectious period PI(j). Due to set-up time, we assume that during the first day of

24

vaccination V(1) = 200,000 only, compared to V(t) = 500,000 t > 1. In the base case,

α =10-7, which implies that R0 = 3. During the first wave of infection, when the epidemic

has not been detected yet, one may expect a higher infection rate. Thus, we assume that

during the first wave R0 = 4. The lack of situational awareness leads also to an extended

stage I period during the first wave. Therefore, we assume that the mean time of stage I

during the first wave is E(I) = 4 days, compared to three days during the rest of the

epidemic period. Table 4.4 presents the effect of these initial conditions on the number of

infected—in comparison to the base case.

Scenario PVP MVP Base Case 1,015 2,232

Initial Conditions 2,131 4,218 Base Case & E(I) = 4 1,814 3,548 Base Case & R0 = 4 1,052 2,415

Base Case & V(1) = 200K 1,083 3,415

Table 4.4: The Effect of Initial Conditions.

The effect of the initial conditions is not negligible. They result in almost 90% more

casualties in MVP and twice that number in PVP. It is also observed that the length of the

infectious period (Stage I) distribution function has the largest single impact among the

three factors that affect the initial conditions. It follows that early pre-symptomatic

detection of the epidemic is crucial in any vaccination policy.

5. Summary and Conclusions

We have developed a new difference-equation model that captures key

epidemiological and operational features of a bioattack response process. The model

represents several inter-temporal parameters and processes—in particular, the process in

which infected individuals become potentially traceable. The model has been

implemented to analyze three vaccination policies: the mass vaccination process (MVP),

the trace vaccination process (TVP), and the newly suggested prioritized vaccination

process (PVP), in which high-priority tracing is conducted in conjunction with a

complement mass vaccination effort.

25

The first conclusion of the analysis is a confirmation of the general result in [4] and

[5], that is, mass vaccination is far more effective than trace vaccination. The second

conclusion is that the prioritized vaccination policy is superior to the mass vaccination

policy for any set of realistic parameters. Moreover, since the PVP “wastes” vaccination

resources on tracing, one may argue that the tradeoff between MVP and PVP may be

sensitive to the assumption regarding the service tracing service reduction factor (c) that

reflects the degradation in the vaccination rate due to tracing. It is shown that this is not

the case; PVP is more effective than MVP even if this ratio is higher than 60. It is also

noted that the advantage of PVP over MVP increases as the vaccination resources become

more limited (see Figure 4.5). The effectiveness of the PVP is also relatively insensitive

to the size of the index set M. Tracing as low as five individuals per index case may be

sufficient for obtaining satisfactory results. Finally, it is noted that initial awareness to

such an attack, which may reduce the length of the first generation infectious stage (I),

can have a significant effect on the total number of infected individuals. From the

logistical point of view, the maximum daily isolation capacity that is needed for MVP is

considerably higher than the capacity needed for PVP.

Recall that the model presented here assumes homogeneous mixing. This assumption

may not be realistic in many possible scenarios. Future work in response-policy analysis

must take into account spatial and social effects. The newly emerging concept of

“small world” network [18] may be utilized to model these effects.

Acknowledgement: I would like to thank Professors Ed Kaplan, Rick Rosenthal, and

Al Washburn for their valuable comments on earlier drafts of this paper.

26

REFERENCES [1] Huerta, M., Balicer, R.D., and Leventhal, A., “SWOT Analysis: Strengths,

Weaknesses, Opportunities and Threats of the Israeli Smallpox Revaccination Program,”

Israel Medical Association Journal, 5 (2003), pp. 42-46.

[2] Broad, W.J., “Study Uses Math Model to Determine Effects of a Smallpox Attack,”

New York Times, July 8, 2002.

[3] Healy, B., “Time for Pause,” US News & World Report, April 21, 2003.

[4] Kaplan, E.H., Craft, D.L., and Wein, L.M., “Emergency Response to a Smallpox

Attack: The Case for Mass Vaccination,” Proceedings of the National Academy of

Science, 99 (16) (2002), pp. 10935-10940.

[5] Kaplan, E.H., Craft, D.L., and Wein, L.M., “Analyzing bioterror response logistics:

the case of smallpox,” Mathematical Biosciences, 183 (2003), pp. 33-72.

[6] Halloran, M.E., Longini, Jr., I.M., Azhar, N., and Yang, Y., “Containing Bioterrorist

Smallpox", Science, 298 (2002), 1428-1432.

[7] Kaplan, E.H., Wein, L.M.; Halloran, M.E., Longini, I.M., “Smallpox Bioterror

Response,” Letter to the Editor, Science, 300 (2003), pp. 1503-1504.

[8] Meltzer, M.I., Damon, I., LeDuc, J.W., and Millar, J.D., “Modeling Potential

Responses to Smallpox as a Bioterrorist Weapon,” Emerging Infectious Diseases, 7 (6)

(2001), pp. 959-969.

[9] Koopman, J., “Controlling Smallpox,” Science 298 (2002), pp. 1342-1344.

27

[10] Müller, J., Kretzschmar, M., and Dietz, K., “Contact Tracing in Stochastic and

Deterministic Epidemic Model,” Mathematical Biosciences, 164 (2000), pp. 39-64.

[11] Newman, M.E.J., “Spread of Epidemic Disease on Networks,” Physical Review,

E 66 (2002), pp. 0161281-01612811.

[12] Müller, J., Schönfisch, B., and Kirkilionis, M., “Ring Vaccination,” J. Mathematical

Biology, 41 (2000), pp. 143-171.

[13] Rhodes, C.J., and Anderson, R.M., “Epidemic Threshold and Vaccination in a

Lattice Model of Disease Spread,” Theoretical Population Biology, 52 (1997),

pp. 101-118.

[14] Epstein, J.M., Cummings, D.A.T., Chakravarty, S., Singa, R.M., and Burke, D.S.,

“Toward a Containment Strategy for Smallpox Bioterror: An Individual-Based

Computational Approach,” Brookings Institution – Johns Hopkins University Center on

Social and Economic Dynamics, Working Paper No. 31 (2002).

[15] Bozzette, S.A., Boer, R., Bhatnagar, V., Brower, J.L., Keeler, E.B., Morton, S.C.,

and Stoto, M.A., “A Model for a Smallpox-Vaccination Policy,” The New England

Journal of Medicine, 348 (5) (2003), pp. 416-425.

[16] Fenner, F., Henderson, D.A., Arita I., Jexek, Z., and Lanyi, I.D., “Smallpox and its

Eradication,” World Health Organization (WHO), Geneva, (1988).

[17] Singh, S., “Some Aspects of the Epidemiology of Smallpox in Nepal,” World Health

Organization (WHO), SE/69.10, Geneva (1969).

[18] Watts, D.J., and Strogatz, S.H., “Collective Dynamics of 'Small World' Networks,”

Nature, 393 (4) (June 1998), pp. 440-442.

28

INITIAL DISTRIBUTION LIST

1. Research Office (Code 09).............................................................................................1 Naval Postgraduate School Monterey, CA 93943-5000

2. Dudley Knox Library (Code 013)..................................................................................2 Naval Postgraduate School Monterey, CA 93943-5002

3. Defense Technical Information Center ..........................................................................2 8725 John J. Kingman Rd., STE 0944 Ft. Belvoir, VA 22060-6218

4. Richard Mastowski (Editorial Assistant) .......................................................................2 Department of Operations Research Naval Postgraduate School Monterey, CA 93943-5000

5. Professor Moshe Kress ..................................................................................................1 Department of Operations Research Monterey, CA 93943-5000

6. Dr. Michael Senglaub ....................................................................................................1 Sandia National Laboratories P.O. Box 5800 MS 0785 Albuquerque, NM 87185

7. Professor Lawrence M. Wein ........................................................................................1 Graduate School of Business Stanford University 518 Memorial Way Stanford, CA 94305-5015

8. Professor Edward H. Kaplan..........................................................................................1 Yale School of Management Box 208200 New Haven, CT 06520-8200

9. Professor Manfred Green...............................................................................................1 ICDC Gertner Institute Haim Sheba Medical Center, Tel Hashomer 52621 Israel

29

10. Professor Gennady Samorodnitsky................................................................................1 School of Operations Research and Industrial Engineering Cornell University Rhodes Hall Ithaca, NY 148

11. Professor Alok R. Chaturvedi ........................................................................................1 Krannert School of Mangement Purdue University West Lafayette, IN 47907

12. Dr. Moti Weiss...............................................................................................................1 CEMA P.O. Box 2250 (T1) Haifa 31021 Israel

13. Distinguished Professor Donald P. Gaver .....................................................................1 Department of Operations Research Naval Postgraduate School Monterey, CA 93943-5000

14. Professor Patricia A. Jacobs...........................................................................................1 Department of Operations Research Naval Postgraduate School Monterey, CA 93943-5000

15. Professor Alan R. Washburn..........................................................................................1 Department of Operations Research Naval Postgraduate School Monterey, CA 93943-5000

16. Professor Richard Rosenthal..........................................................................................1 Department of Operations Research Naval Postgraduate School Monterey, CA 93943-5000

17. Professor Ted Lewis ......................................................................................................1 Department of Computer Science Naval Postgraduate School Monterey, CA 93943-5000

18. Professor James Fobes ...................................................................................................1 Department of Aeronautics and Astronautics Naval Postgraduate School Monterey, CA 93943-5000

30

19. Professor Dan Dolk........................................................................................................1 Department of Information Sciences Naval Postgraduate School Monterey, CA 93943-5000